Basic and degenerate pregeometries

Abstract

We study pairs $(Γ,G)$, where $Γ$ is a 'Buekenhout-Tits' pregeometry with all rank 2 truncations connected, and $G\leqslant\mathrm{Aut} Γ$ is transitive on the set of elements of each type. The family of such pairs is closed under forming quotients with respect to $G$-invariant type-refining partitions of the element set of $Γ$. We identify the 'basic' pairs (those that admit no non-degenerate quotients), and show, by studying quotients and direct decompositions, that the study of basic pregeometries reduces to examining those where the group $G$ is faithful and primitive on the set of elements of each type. We also study the special case of normal quotients, where we take quotients with respect to the orbits of a normal subgroup of $G$. There is a similar reduction for normal-basic pregeometries to those where $G$ is faithful and quasiprimitive on the set of elements of each type.